Solve 4*12=e^-2x

$$4 \cdot 12 = e ^ { - 2 x }$$

Solve for x

$$x=-\frac{\ln(3)}{2}-2\ln(2)\approx -1.935600505$$

Steps Using Definition of Logarithm

Multiply 4 and 12 to get 48.

$$48=e^{-2x}$$

Swap sides so that all variable terms are on the left hand side.

$$e^{-2x}=48$$

Take the logarithm of both sides of the equation.

$$\log(e^{-2x})=\log(48)$$

The logarithm of a number raised to a power is the power times the logarithm of the number.

$$-2x\log(e)=\log(48)$$

Divide both sides by \log(e).

$$-2x=\frac{\log(48)}{\log(e)}$$

By the change-of-base formula \frac{\log(a)}{\log(b)}=\log_{b}\left(a\right).

$$-2x=\log_{e}\left(48\right)$$

Divide both sides by -2.

$$x=\frac{\ln(48)}{-2}$$

Solve for x (complex solution)

$$x=-i\pi n_{1}-\frac{\ln(48)}{2}$$

$$n_{1}\in \mathrm{Z}$$